Should you give her your number instead of asking for hers?

Should you give her your number instead of asking for hers? Sometimes, yes. It hands her the choice, which suits a few situations well. It also gives up your ability to follow up, which is a real cost. Neither option is the right default for everyone, so here are the tradeoffs and some lines for when you do it.

What changes when you give yours

When you ask for her number, you hold the next step. You can text first, and the follow-up is in your hands. When you give yours, she holds it. She decides whether, when and how to message you, and you wait.

That's the whole difference. Everything else follows from it.

What it has going for it

It's lower pressure for her. She doesn't have to say yes or no to handing out a number. She can take yours, say thanks, and decide later. Some people prefer to share their number only once they know someone better, and giving yours respects that.
It can read as confident and clear. You're saying you'd like to hear from her and leaving it there.
Some people feel safer this way. Not everyone likes to give a stranger a direct line, and this avoids the question.
It works when you're short on time. Her train's pulling in or her friends are waiting, and a quick offer fits in five seconds.
It's easier on your nerves. For some people, the ask is the hardest part. Offering is gentler, and a shy person might do it more often.

What it costs

You may never hear from her. Taking your number doesn't mean she'll text it. Maybe she meant to and forgot, maybe she lost the slip of paper, maybe she wasn't interested. You can't tell which.
You can't follow up. You have no number to text, and tracking her down another way isn't an option if she hasn't written. Any momentum from the conversation now depends on her.
It gives you less information. If she gives you her number when you ask, that's a small yes. Taking yours can be a weaker signal, because it's polite and costs her nothing.
You wait. Waiting by your phone for a day or a week is easy to turn into a mood.
It can come off as passive if you use it to avoid asking, rather than because it fits.

A rule of thumb: treat the number as handed off the moment she takes it, and plan your week as if no text is coming. If one arrives, good.

When it fits

She seems unsure about sharing hers, but hasn't said no and has been engaged: asking you questions, staying, smiling. Offering yours once, with "No pressure, here's mine if you want it," gives her an easy yes. If she says she'd rather not, drop it.
You'll see her again at a class, gym, club or job. You can follow up in person, which makes waiting less costly.
She's rushed. Quick exchange, no pressure.
You're in a setting where asking would put her on the spot, like in front of her group or at her workplace.
You'd prefer she holds the choice. That's a valid reason by itself.

When it doesn't

She said no or "I don't give out my number." Don't turn that into "then take mine." It's the same ask in reverse, and a no is a no. She didn't give you her number: how to take it covers what to do then.
You want to set a date this week, and a gap of unknown length doesn't suit you.
You haven't picked up any signs of interest. There's nothing to wait for. How to tell if someone is interested in you lists what to look for.
You're using it to avoid risk. If you want her number, the clearer move is asking. How to ask for a number has the exact steps.

Exact lines

Short, warm, and with an easy way for her to say no.

The simple offer: "I've really enjoyed talking. If you ever want to grab coffee, here's my number. No pressure either way."
Give her the choice: "Do you want to swap numbers? Or I can give you mine and you can text me if you feel like it."
With a plan attached: "There's a good taco place on Fifth I want to take you to. Put my number in your phone and text me if you're in."
If she's rushed: "I know you've got to go. Want to swap numbers, or should I just give you mine?"
If she seems unsure, once: "No pressure at all. Here's mine if you feel like it." If she says no or waves it off, say "No problem" and leave it.

How to hand it over: say your first name, then either let her type your number into her own phone or write your name and number on a card or receipt. A number with no name attached is easy to ignore.

If she texts, save her number and reply. If she doesn't, leave it.

A middle option

You can combine both. Ask for hers, and give yours at the same time: "Want to swap numbers?" That puts the choice on the table without taking it away. If she gives hers, you can text first. If she'd rather take yours, you're in the offering case.

You can offer your Instagram handle the same way. From Instagram DMs to her number covers moving from a lower-commitment channel to a number.

What happens after

Then you do what you'd do anyway: wait and live your week. If she texts, answer within the same day, mention where you met, and propose a specific plan fairly quickly. What to text after you get a number applies the same way. If she doesn't text, take it as an answer for now. Don't nudge. She has your number and knows where you met. She gave you a fake number covers a related situation.

What to do next

Decide your default, knowing both tradeoffs: ask, offer, or "swap." Write down which you'd pick in three kinds of situation (rushed, hesitant, repeat setting).
Write two offer lines from the list above and say each aloud five times.
Put a day and a plan on the first offer you make: "coffee Saturday" beats "if you ever want to hang out."
After you hand yours over, plan your week as if no text is coming. Make two other plans.
If she texts, reply the same day with where you met and a specific idea.
If nothing arrives in a week, let it go. No second route to reach her.

In Run Game, you can add a note to the log about which route you used, ask or offer. After a month, your notes show how often each route led somewhere.